A Course in Constructive Algebra

Mines, Ray; Richman, Fred; Ruitenburg, Wim

doi:10.1007/978-1-4419-8640-5

Cited by 240 publications

(276 citation statements)

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“…Although the proof we give is constructive, we will not use the "constructive" vocabulary as in [9] or [10] in order to lighten the exposition. For instance, in Lemma 2, for the purpose of obtaining a "fully" constructive formulation, one has to replace "degree" by "formal degree".…”

Section: Conjecture 2 If R Is a Commutative Ring And Vmentioning

confidence: 99%

Stably free modules over $\mathbf{R}[X]$ of rank $> \dim\mathbf{R}$ are free

Yengui

2011

Math. Comp.

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Section: Conjecture 2 If R Is a Commutative Ring And Vmentioning

confidence: 99%

Stably free modules over $\mathbf{R}[X]$ of rank $> \dim\mathbf{R}$ are free

Yengui

2011

Math. Comp.

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“…The chosen scripts constitute a development of constructive algebra [7]à la Bishop [4], up to valued group lattices. It includes 152 theorems and definitions, comprising: constructive setoids (C, =) with the induced equality =; constructive partially ordered set (C, ) with the induced order relations ≤ and <; groups (C, =, +, O, −); lattices (C, ∧, ∨); valuations (K, C, µ); topological spaces (X, O); and all the algebraic structures (such as group lattices) that inherit from them.…”

Section: Performance Of Spurious Error Detectionmentioning

confidence: 99%

Spurious Disambiguation Errors and How to Get Rid of Them

Coen

Zacchiroli

2008

Math.Comput.Sci.

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Abstract. The disambiguation approach to the input of formulae enables users of mathematical assistants to type correct formulae in a terse syntax close to the usual ambiguous mathematical notation. When it comes to incorrect formulae however, far too many typing errors are generated; among them we want to present only errors related to the formula interpretation meant by the user, hiding errors related to other interpretations.We study disambiguation errors and how to classify them into the spurious and genuine error classes. To this end we give a general presentation of the classes of disambiguation algorithms and efficient disambiguation algorithms. We also quantitatively assess the quality of the presented error classification criteria benchmarking them in the setting of a formal development of constructive algebra.

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“…We interpret the previous argument in the framework of constructive mathematics [6,7]. Let L(R) be the lattice of radically finitely generated ideals of R: that is, the radicals of finitely generated ideals [2].…”

Section: Constructive Interpretationmentioning

confidence: 99%

A nilregular element property

2005

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Abstract. An element a of a commutative ring R is nilregular if and only if x is nilpotent whenever ax is nilpotent. More generally, an ideal I of R is nilregular if and only if x is nilpotent whenever ax is nilpotent for all a ∈ I. We give a direct proof that if R is Noetherian, then every nilregular ideal contains a nilregular element. In constructive mathematics, this proof can then be seen as an algorithm to produce nilregular elements of nilregular ideals whenever R is coherent, Noetherian, and discrete. As an application, we give a constructive proof of the Eisenbud-Evans-Storch theorem that every algebraic set in n-dimensional affine space is the intersection of n hypersurfaces.AMS Classification: 03F65 (14M10) The nilregular element propertyLet R be a commutative ring with unit and N its nilradical, i.e. the ideal consisting of its nilpotent elements. We define an element a (respectively, an ideal I) of R to be nilregular if and only if x ∈ N whenever ax ∈ N (respectively, ax ∈ N for all a ∈ I). So an ideal I is nilregular precisely when the transporter ideal (N : I) = {x ∈ R : xI ⊆ N } is contained in N . We present a method to find nilregular elements of nilregular ideals when R is Noetherian. For this, we interpret first the property of being nilregular in a topological way.As usual, let D(a) be the set of prime ideals p of R such that a / ∈ p, and let D(a 1 , . . . , a n ) stand for the union of D(a 1 ), . . . , D(a n ). The intersection of D(a) and D(b) is D(ab), and D(a) is a subset of D(a 1 , . . . , a n ) if and only if a belongs to the radical of the ideal (a 1 , . . . , a n ) generated by a 1 , . . . , a n . In particular, D(a) = ∅ precisely when a ∈ N .It is well-known that the D(a) with a ∈ R form a basis of opens for the Zariski topology on the prime spectrum (the set of all prime ideals) of R. It follows that a ∈ R is nilregular if and only if D(a) is dense for the Zariski topology. Remark 1.2. D(a 1 , . . . , a n ) is dense if and only if (a 1 , . . . , a n ) is a nilregular ideal. Theorem 1.3. Let R be Noetherian, and a 1 , . . . , a n ∈ R. If D(a 1 , . . . , a n ) is dense, then the ideal (a 1 , . . . , a n ) contains a nilregular element.Proof. If D(x) = ∅, then there exists i such that D(xa i ) = ∅, because D(a 1 , . . . , a n ) is dense. Hence if the ring is nontrivial, then we can inductively build a sequence b 0 , b 1 , . . .

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A Course in Constructive Algebra

Cited by 240 publications

References 0 publications

Stably free modules over $\mathbf{R}[X]$ of rank $> \dim\mathbf{R}$ are free

Stably free modules over $\mathbf{R}[X]$ of rank $> \dim\mathbf{R}$ are free

Spurious Disambiguation Errors and How to Get Rid of Them

A nilregular element property

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